I'm not from Wyoming, but I just spent last week driving around the State camping and hiking. First off this is a hot year for bears in Wyoming. They're seeing record numbers in all areas; they even had one in downtown Dubois one evening. Your best bet would be to drive southeast from Jackson Hole, that'll put you in the desert in an hour or two. As to exactly where to hunt, they have these map booklets that are the size of one of those Rand McNalley atlases of the States (these book's are difficult to find, you're gonna have to look around - but downtown Jackson would be a good place to start). These book's have very concice maps of each county which also contain color coordinated ownership of the land on them. Purple for BLM, white for private, ect....
No need to obtain a license if you're just hunting coyotes or prarie dogs - just can't hunt in the National Parks. Good luck. Let me know if you have any more questions.